An ensemble density matrix model that includes one- and two-body losses is derived for a trapped-atom clock. A trapped-atom clock is mainly affected by one- and two-body losses, generally giving nonexponential decays of populations; nevertheless, three-body recombination is also quantitatively analyzed to demonstrate the boundaries of its practical relevance. The importance of one-body losses is highlighted without which population trapping behavior would be observed. The model is written with decay constants expressed through experimental parameters. It can complement, e.g., the ISRE (identical spin rotation effect) model to improve its predictions: ISRE dramatically increases the ensemble coherence time, hence it enables one to observe the influence of two-body losses on the interferometry contrast envelope. The presented model is useful for Ramsey interferometry and is ready for immediate experimental verification in existing systems.
An ensemble density matrix model that includes one- and two-body losses is derived for a trapped-atom clock. A trapped-atom clock is mainly affected by one- and two-body losses, generally giving nonexponential decays of populations; nevertheless, three-body recombination is also quantitatively analyzed to demonstrate the boundaries of its practical relevance. The importance of one-body losses is highlighted without which population trapping behavior would be observed. The model is written with decay constants expressed through experimental parameters. It can complement, e.g., the ISRE (identical spin rotation effect) model to improve its predictions: ISRE dramatically increases the ensemble coherence time, hence it enables one to observe the influence of two-body losses on the interferometry contrast envelope. The presented model is useful for Ramsey interferometry and is ready for immediate experimental verification in existing systems.
We present a number of analytical means for the analysis of the evolution of an atom clock, emphasizing the role of many-body processes in the population and coherence loss. A master equation with one-body and two-body losses included is presented for a thermal cloud ensemble, suitable for modelling atom clock evolution. Approximate solutions for the coupled rate equations with one-body and two-body losses are obtained for the case of Ramsey free evolution, and their validity is analyzed. An approach is suggested to combine the analytical trap-induced fringe envelope function with a numerically evaluated many-body envelope function. A set of Ramsey and spin echo models is offered with analytical solutions for the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. the off-resonant and on-resonant regimes with one-body population loss and phenomenological dephasing. Their detuning spectra are presented and the difference between them and the standard Ramsey model is shown: the Rabi pedestal in the present model has no spectral ripples due to the pi/2-pulse duration equal to a quarter of a Rabi oscillation, even off resonance, in contrast to the Ramsey method where the pi/2-pulse duration is not corrected off resonance and causes spectral ripples. An interferometric fringe in long evolution experiments is typically measured with a visibility decay primarily due to phase diffusion, phase instability, trapping effects and many-body losses. It constitutes a problem to extract parameters from a fringe obtained in the time domain because of the growing uncertainty of the fringe amplitude and phase. This visibility loss can be compensated for systems with unequal state population losses, as the one we have, by tailoring the pulse durations. Methods are suggested for the one-body and many-body cases.A comprehensive set of experiments is conducted on the characterization of the stability of the trapped atom clock: with condensate and thermal cloud ensembles, with phase-domain Ramsey and spin echo interferometries, with synchronization from a rubidium frequency standard and from an oven-controlled quartz oscillator. The types of noise present in the system are identified by means of an Allan deviation analysis. Despite the 50 times different cloud densities of the condensate and the thermal clouds, an equality of the phase stability in the Ramsey interferometry with BEC and thermal clouds is demonstrated. The spin echo interferometry has shown a slightly superior stability with the thermal clouds as compared to the BEC.
We offer interferometry models for thermal ensembles with one-body losses and the phenomenological inclusion of perturbations covering most of the thermal atom experiments. A possible extension to the many-body case is briefly discussed. The Ramsey pulses are assumed to have variable durations and the detuning during the pulses is distinguished from the detuning during evolution. Consequently, the pulses are not restricted to resonant operation and give more flexibility to optimize the interferometer to particular experimental conditions. On this basis another model is devised in which the contrast loss due to the unequal one-body population decays is canceled by the application of a nonstandard splitting pulse. For the importance of its practical implications, an analogous spin-echo model is also provided. The developed models are suitable for the analysis of atomic clocks and a broad range of sensing applications; they are particularly useful for trapped-atom interferometers.
Achieving precise control of ultracold atomic gases requires a detailed understanding of atom loss mechanisms. Motivated by the anomalous three-body decay in a three-component Fermi gas reported in Ref. [1], this work investigates mechanisms that possibly contribute to the observed loss. The three-body Schrödinger equation is solved in the hyperspherical adiabatic representation with pairwise van der Waals interactions, and the S-matrix is obtained via the eigenchannel R-matrix method to compute recombination rate coefficients K3 and two-body cross sections. At the magnetic field strength where the anomalous decay occurs, K3 is unitary limited, exhibiting the threshold energy scaling K3(E)∝E−1. Consequently, the thermally averaged ⟨K3⟩ acquires a temperature dependence. Because the experiment is performed in the degenerate regime, ⟨K3⟩ also explicitly depends on the per-spin densities through the per-spin Fermi energies EF(i)∝ni2/3. As the gas is diluted and degeneracy is reduced, ⟨K3⟩ approaches the non-degenerate value and becomes a function of temperature only. Channel-resolved branching ratios and cross sections are folded into a Monte Carlo cascade simulation of secondary collisions and trap escape. The analysis indicates that typical three-body recombination events remove fewer than three atoms on average, and that the atom losses are primarily due to the ejection of secondary collision products, rather than the initial three-body recombination products. Therefore, a significant fraction of the released binding energy remains in the trapped ensemble as kinetic energy. Retained energy drives evaporative loss, offering a plausible, partial explanation for the anomalous decay.
This thesis describes new results on the entanglement of atomic spins in Ramsey interferometry, optical atomic clocks and trapped ions. It is divided into three parts: First, we investigate improvements to conventional Ramsey interferometry with entanglement and adding only rotations of the collective spin to adjust the signal and measurement directions. The geometric degrees of freedom, connected to the rotations, are analytically optimized for a large class of generalized Ramsey protocols to allow efficient optimization of all parameters. Besides a unification of existing approaches, the main result is that there is only one new protocol, where a previously unused double inversion is applied. Studies of the local sensitivity show that this protocol reaches the fundamental quantum Fisher information limit and is yet robust against errors during preparation and measurement. In the second section we investigate the conditions under which optical atomic clocks exhibit increased long-term stability when applying weakly entangled, spin squeezed states. We discuss the common case of an atomic clock with a single ensemble, typical Brownian frequency noise and finite dead time. Theoretical modelling of the servo loop allows quantitative predictions of the optimal stability for given values of dead time and laser noise, in very good agreement with numerical simulations of the closed feedback loop. The main result is that, even with the current most stable lasers, the clock stability can only be improved for ensembles below a critical atom number of about one thousand in optical Sr lattice clocks. Even with a future improvement of the laser performance by one order of magnitude, the critical atom number still remains below 100,000. In contrast, clocks based on smaller, non-scalable ensembles, such as ion clocks, can already benefit from squeezed states with current clock lasers. Thus the last section considers the robust generation of entanglement in ion traps. An error budget including relevant experimental error sources is calculated for state-of-the-art quantum gates, driven by oscillating microwave gradients in surface traps. Amplitude modulation of the driving fields is shown to efficiently counteract the current limitations from motional mode instability. The predicted increase of the gate quality was demonstrated by the group of C. Ospelkaus at PTB Braunschweig, who measured gates with errors as low as ~ 10^(-3). In a similar approach, interactions between spin and motion can also be generated by combining oscillating rf-fields with a static magnetic field gradient. Penning traps designed for precision spectroscopy already feature large magnetic field gradients at the edge of a magnetic bottle configuration. We present parameters and conditions under which laser-free coupling of spin and quantized motion for (anti-)protons is possible at these points, in a step towards quantum logic spectroscopy for (anti-)protons.
We experimentally and theoretically study phase coherence in two-component Bose-Einstein condensates of ^{87}Rb atoms on an atom chip. Using Ramsey interferometry we determine the temporal decay of coherence between the |F=1,m_{F}=-1⟩ and |F=2,m_{F}=+1⟩ hyperfine ground states. We observe that the coherence is limited by random collisional phase shifts due to the stochastic nature of atom loss. The mechanism is confirmed quantitatively by a quantum trajectory method based on a master equation which takes into account collisional interactions, atom number fluctuations, and losses in the system. This decoherence process can be slowed down by reducing the density of the condensate. Our findings are relevant for experiments on quantum metrology and many-particle entanglement with Bose-Einstein condensates and the development of chip-based atomic clocks.
We implement coherent delocalization as a tool for improving the two primary metrics of atomic clock performance: systematic uncertainty and instability. By decreasing atomic density with coherent delocalization, we suppress cold-collision shifts and two-body losses. Atom loss attributed to Landau-Zener tunneling in the ground lattice band would compromise coherent delocalization at low trap depths for our ^{171}Yb atoms; hence, we implement for the first time delocalization in excited lattice bands. Doing so increases the spatial distribution of atoms trapped in the vertically oriented optical lattice by ∼7 times. At the same time, we observe a reduction of the cold-collision shift by 6.5(8) times, while also making inelastic two-body loss negligible. With these advantages, we measure the trap-light-induced quenching rate and natural lifetime of the ^{3}P_{0} excited state as 5.7(7)×10^{-4} E_{r}^{-1} s^{-1} and 19(2) s, respectively.